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Vector Analysis : Construction of a Plane Curve

Please see the attached file for the fully formatted problems.

Problem: Assume you are given a non-negative function K(s). We would like to construct a plane curve B(s) with curvature
K(s).....

HINT: Use the Fundamental Theorem of Calculus to show B has unit speed and then compute dT/ds.

Problem: I. If K(s) = ... using the above datum, construct a formula for B(s).

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Problem:
Assume you are given a non-negative function (s). We would like to construct a plane curve (s) with curvature (s).
Show that such a  may be defined by

where

Hint: Use the fundamental theorem of Calculus to show  has a unit speed and then compute dT/ds.
If , using the above datum, construct a formula for (s).

Solution:
First of all, we need to point out that (s) is the vector whose components are the coordinates of a current point of the curve:
( 1)
which is defined parametrically. As the parameter (s) is varying let's say from ...

Solution Summary

Construction of a Plane Curve is investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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